The primary distinction is the part of speech. The meaning is the same. I can use either word in that sentence. However, I can start a sentence with however, but I need but if I need it to be a conjunction. I can start a sentence with but. However, I needed to make this a new sentence, since I used however, but if I had used but, I could have continued the previous sentence. But that won't matter if either word starts a sentence.
When though appears at the beginning it is synonymous with although and heads up a subordinate clause, which makes English speakers expect a main clause. For example: I did not see him, though (or although) he was there. Although (or though) he was there, I did not see him.
With a final "though", you can get two separate sentences. I did not see him yesterday. He was there, though.
It can also appear in an answer to a question or statement, but even then you will a main clause in the response. "Was he there yesterday?" "Yes, he was, though I didn't see him". This main clause can be reduced to a simple "Yes", but it will remain.
A: "Was he there yesterday?" B: "Yes, though I didn't see him."
A: "He was there yesterday." B: "Yes, though I did not see him."
That, perhaps, could be correct, but your "however" should be be fully set off with commas: "That is not, however
, the case."
It may be that die Eule wants the "not" to be more closely tied to object (the case) than the verb (ist). Personally, I don't know that it makes that much difference.
Both " . . . not correct" and " . . . not true" are not as faithful a translation as " . . . . not like that" or " . . . not that way" or " . . . not the case" or even " . . . not thus[ly]".
While "so" (English) can be used as an adjective meaning "true" or "correct", the German "so" does not have that usage. In this sentence, so is an adverb meaning "thus, like this/that, in this/that way, in this/that manner". (Ref: Wiktionary and Duden).